via the definition of
\(c_{1}^{\mathrm {loc}}(L_{\varrho },s_{\varrho }): A_*(\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}})\rightarrow A_{*-1}(\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho )\) 
as the refined Gysin homomorphism \(0_{\pi _{P}^*L_{\varrho }}^!\)  of some normal bundle as in [[1]}, Section 3.1] through the diagram
\({5pc}{\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho [d] [r] & \mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}} [d]^-{\pi _{P}^*s_{\varrho }} \\\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}[r]^-{0_{\pi _{P}^*L_{\varrho }}} & \pi _{P}^*L_{\varrho } &}\) 
